Local search is a powerful heuristic in optimization and computer science, the complexity of which was studied in the white box and black box models. In the black box model, we are given a graph $G = (V,E)$ and oracle access to a function $f : V \to \mathbb{R}$. The local search problem is to find a vertex $v$ that is a local minimum, i.e. with $f(v) \leq f(u)$ for all $(u,v) \in E$, using as few queries as possible. The query complexity is well understood on the grid and the hypercube, but much less is known beyond. We show the query complexity of local search on $d$-regular expanders with constant degree is $\Omega\left(\frac{\sqrt{n}}{\log{n}}\right)$, where $n$ is the number of vertices. This matches within a logarithmic factor the upper bound of $O(\sqrt{n})$ for constant degree graphs from Aldous (1983), implying that steepest descent with a warm start is an essentially optimal algorithm for expanders. The best lower bound known from prior work was $\Omega\left(\frac{\sqrt[8]{n}}{\log{n}}\right)$, shown by Santha and Szegedy (2004) for quantum and randomized algorithms. We obtain this result by considering a broader framework of graph features such as vertex congestion and separation number. We show that for each graph, the randomized query complexity of local search is $\Omega\left(\frac{n^{1.5}}{g}\right)$, where $g$ is the vertex congestion of the graph; and $\Omega\left(\sqrt[4]{\frac{s}{\Delta}}\right)$, where $s$ is the separation number and $\Delta$ is the maximum degree. For separation number the previous bound was $\Omega\left(\sqrt[8]{\frac{s}{\Delta}} /\log{n}\right)$, given by Santha and Szegedy for quantum and randomized algorithms. We also show a variant of the relational adversary method from Aaronson (2006), which is asymptotically at least as strong as the version in Aaronson (2006) for all randomized algorithms and strictly stronger for some problems.

翻译：局部搜索是优化和计算机科学中一种强大的启发式方法，其复杂度已在白盒和黑盒模型中得到研究。在黑盒模型中，我们给定图$G = (V,E)$和函数$f: V \to \mathbb{R}$的预言机访问权限。局部搜索问题是在尽可能少的查询次数内找到局部最小值顶点$v$，即对于所有$(u,v) \in E$，满足$f(v) \leq f(u)$。该问题在网格和超立方体上的查询复杂度已被充分理解，但在其他结构上的认知较少。我们证明，在常度数$d$-正则展开图上，局部搜索的查询复杂度为$\Omega\left(\frac{\sqrt{n}}{\log{n}}\right)$，其中$n$为顶点数。这一结果与Aldous (1983)针对常度数图给出的上界$O(\sqrt{n})$仅相差对数因子，表明带热启动的最速下降法本质上是对展开图的最优算法。此前已知的最佳下界为$\Omega\left(\frac{\sqrt[8]{n}}{\log{n}}\right)$，由Santha和Szegedy (2004)针对量子和随机算法给出。我们通过考虑更广泛的图特征（如顶点拥塞度和分离数）获得该结果。我们证明，对于每个图，局部搜索的随机查询复杂度为$\Omega\left(\frac{n^{1.5}}{g}\right)$，其中$g$为图的顶点拥塞度；以及$\Omega\left(\sqrt[4]{\frac{s}{\Delta}}\right)$，其中$s$为分离数，$\Delta$为最大度数。对于分离数，此前由Santha和Szegedy针对量子和随机算法给出的界为$\Omega\left(\sqrt[8]{\frac{s}{\Delta}} / \log{n}\right)$。我们还给出了Aaronson (2006)中关系对抗方法的一个变体，该变体在渐近意义下至少与Aaronson (2006)版本对所有随机算法具有相同强度，并对某些问题严格更强。